Acoustic Waves Part 1 pdf

Acoustic Waves edited by Don W. Dissanayake SC I YO Acoustic Waves Edited by Don W. Dissanayake Published by Sciyo Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2010 Sciyo All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by Sciyo, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Ana Nikolic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Marco Rullkoetter, 2010. Used under license from Shutterstock.com First published September 2010 Printed in India A free online edition of this book is available at www.sciyo.com Additional hard copies can be obtained from publication@sciyo.com Acoustic Waves, Edited by Don W. Dissanayake p. cm. ISBN 978-953-307-111-4 SC I YO. C O M WHERE KNOWLEDGE IS FREE free online editions of Sciyo Books, Journals and Videos can be found at www.sciyo.com [...]... Solids W = W1( ) [ 1 ] ⊕ W2( ) [φ2 ] ⊕ W3( ) [φ3 ] 1 1 1 (11 9) The eigen-qualities and eigen-operators of monoclinic crystal are respectively shown as below * E1 = φ1T ⋅ E = * E2 = φ2T ⋅ E = 1 ( 1 − e 11 ) 2 2 + e12 2 2 + e12 1 (η2 − e 11 ) ⎡( 1 − e 11 ) E1 + e12 E2 ⎦ ⎤ ⎣ (12 0) ⎡ ⎤ ⎣ e12 E1 + (η 2 − e 11 ) E2 ⎦ (12 1) * E3 = φ3T ⋅ E = E3 * 1 =− * 2 =− 2 e12 ( 1 − e 11 ) 2 2 e12 (η2 − e 11 ) 2 (12 2) ⎡⎛ η −... (30) where ϕ 1, 2 = c13 2 ( 1, 2 − c 11 − c 12 ) + 2c13 2 ϕ3 = [1, 1, 0,0,0,0]T, 2 2 × [1, 1, 1, 2 − c 11 − c12 c3 ϕ i = ξi , i = 4, 5,6 ⎫ ,0,0,0]T ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ( 31) The eigenelasticity and eigenoperator of hexagonal solids are 2 ⎫ c 11 + c12 + c 33 ⎛c +c +c ⎞ 2 ± ⎜ 11 12 33 ⎟ + 2c13 , ⎪ ⎪ 2 2 ⎝ ⎠ ⎪ λ3 = c 11 − c12 , λ4 = c 44 , ⎪ ⎪ 2 ⎬ c13 * * 1, 2 = ∇ III , ⎪ 2 2 ⎪ ( 1, 2 − c 11 − c12 ) + 2c13 ⎪ 2 2 1 2 ⎪ * *... ⎣( 1 − e 11 ) E1 + e12 E2 ⎦ ⎤ ⎡ ⎤ ⎣ (12 6) * 2 ⎡ e12 E1 + (η 2 − e 11 ) E2 ⎦ = μ0η 2 ∂ t2 ⎣ e12 E1 + (η 2 − e 11 ) E2 ⎦ ⎤ ⎡ ⎤ ⎣ (12 7) (∂ 2 x ) + ∂ 2 E3 = μ0 e33∂ t2 E3 y (12 8) The velocities of electromagnetic waves are respectively as follows c( ) = 1 1 c( ) = 2 ⎧( e + e ) ⎫ ⎪ 1 ⎤ 2 ⎪ μ0 ⎨ 11 22 + ⎢ ( e 11 − e22 ) ⎥ + e12 ⎬ 2 ⎣2 ⎦ ⎪ ⎪ ⎩ ⎭ 1 2 2 ⎧( e + e ) ⎫ ⎪ 1 ⎤ 2 ⎪ μ0 ⎨ 11 22 − ⎢ ( e 11 − e22 ) ⎥ + e12... ⎢⎜ 1 11 ⎟ ∂ 2 + ∂ 2 + ( ∂ 2 + ∂ 2 ) − 2 ⎜ 1 11 ⎟ ∂ 2 ⎥ z y x z xy 2 + e12 ⎢⎝ e12 ⎠ ⎝ e12 ⎠ ⎥ ⎣ ⎦ (12 3) ⎡⎛ η − e ⎞ 2 ⎛η − e ⎞ ⎤ ⎢⎜ 2 11 ⎟ ( ∂ 2 + ∂ 2 ) + ∂ 2 + ∂ 2 − 2 ⎜ 2 11 ⎟ ∂ 2 ⎥ x z z y xy 2 + e12 ⎢⎝ e12 ⎠ ⎝ e12 ⎠ ⎥ ⎣ ⎦ (12 4) ( ) ( * 3 ( = − ∂2 + ∂2 x y ) ) (12 5) Therefore, the equations of electromagnetic waves in monoclinic crystal can be written as below * 1 ⎡( 1 − e 11 ) E1 + e12 E2 ⎦ = μ0 1 ... (10 7) * 3 = − ∂2 + ∂2 x y ( ) (10 8) Therefore, the equations of electromagnetic waves in biaxial crystal can be written as below (∂ 2 z ) + ∂ 2 E1 = μ0 e 11 t2 E1 y (10 9) 14 Acoustic Waves (∂ 2 x + ∂ 2 ) E2 = μ0 e22 ∂ t2 E2 z (∂ 2 x + ∂ 2 E3 = μ0 e33∂ t2 E3 y ) (11 0) (11 1) The velocities of electromagnetic waves are respectively as follows c( ) = 1 μ0 e 11 (11 2) c( ) = 1 μ0 e22 (11 3) 1 2 c( ) = 3 1. .. ⎢ 2 2 ⎪ ⎦ ( 1 − e 11 ) + e12 ⎣ e12 ⎪ T ⎪ ⎡ η 2 − e 11 ⎤ e12 ⎪ φ2 = 1, ,0 ⎥ ⎨ ⎢ 2 2 ⎦ ⎪ (η2 − e 11 ) + e12 ⎣ e12 ⎪ T ⎪φ3 = [ 0,0 ,1] ⎪ ⎪ ⎩ (11 7) where 1, 2 = ( e 11 + e22 ) ± 2 2 1 ⎤ 2 ⎢ 2 ( e 11 − e22 ) ⎥ + e12 , η 3 = e33 ⎣ ⎦ (11 8) We can see from the above equations that there are also three eigen-spaces in monoclinic crystal, and the space structure is following 15 The Eigen Theory of Waves in Piezoelectric... 2 ∂ 12 , ⎪ 3 2 ⎭ 1, 2 = ( (32) ) Thus there exist four independent elastic waves in hexagonal solids, which can be described by following equations λi Δ i*ε i * ( x , t ) = ρε i * ( x , t ) i = 1, 2, 3 , 4 (33) where * ε 1 ,2 = c13 2 ( 1, 2 − c 11 − c12 ) + 2c13 2 × [ε 11 + ε 22 + ( 1, 2 − c 11 − c12 c13 )ε 33 ] (34) The Eigen Theory of Waves in Piezoelectric Solids * ε3 = 7 1 2 (ε 11 − ε 22 )2 + ε 12 ... symmetry on ( i , j ) in Eq.(9) we can rewrite it in the form of matrix Δσ = ρΔttε (10 ) where ⎡ ∂ 11 ⎢ ⎢0 ⎢0 Δ=⎢ ⎢0 ⎢ ⎢ ∂ 13 ⎢∂ ⎣ 12 0 0 ∂ 22 0 ∂ 33 0 ∂ 23 ∂ 23 0 ∂ 13 ∂ 12 0 0 ∂ 32 ∂ 32 ( ∂ 22 + ∂ 33 ) ∂ 31 0 ∂ 31 ∂ 21 ∂ 12 ( ∂ 11 + ∂ 33 ) ∂ 13 ∂ 23 ∂ 21 ⎤ ⎥ ⎥ ⎥ ⎥ ∂ 31 ⎥ ⎥ ∂ 32 ⎥ ∂ 22 + ∂ 11 ) ⎥ ( ⎦ ∂ 21 0 (11 ) 4 Acoustic Waves It is seen that Δ is a symmetrical differential operator matrix, and ∂ ij... 1 (11 4) μ0 e33 It is seen that there are three kinds of electromagnetic waves in biaxial crystal 3.3.4 Monoclinic crystal The matrix of dielectric permittivity of monoclinic dielectrics is following ⎡ e 11 e12 0 ⎤ ⎢ ⎥ e = ⎢ e12 e22 0 ⎥ ⎢0 0 e33 ⎥ ⎣ ⎦ (11 5) The eigen-values and eigen-vectors are respectively shown as below Γ = diag [ 1 ,η2 , e33 ] (11 6) T ⎧ ⎡ 1 − e 11 ⎤ e12 ⎪ 1 = ,1, 0 ⎥ ⎢ 2 2 ⎪ ⎦ ( 1 −... {Δ } H * I i = 1, 2, ,m (15 5) { } I = 1 n (15 6) { } I =1 n (15 7) = −∇t ϑI* BI* * I = ∇t φI* DI* Substituting Eq . (14 1)- (14 3) into Eq. (15 5)- (15 7), we have {Δ } E * I {Δ } H * I * I = −∇t {ϑI } γ I H I* ( I = 1 n = ∇t {φI } η I EI* + gIjε j* * I ( ) T Δ i* λiε i* − g iJ EJ* = 0 ) (15 8) I =1 n (15 9) i =1 m (16 0) { } Transposing Eq. (15 8), and multiplying it with Δ * , and also using Eq. (15 9), we have I . eigenoperator of hexagonal solids are () () 2 2 11 12 33 11 12 33 1, 2 13 311 12 444 2 ** 13 1, 2 2 2 1, 2 11 12 13 *2* 2 34 12 2, 22 ,, , 2 21 ,2, 32 III II III ccc ccc c cc c c cc c λ λλ Δ λ ΔΔ ⎫ ++. it in the form of matrix. Δ tt ρ Δ = σ ε (10 ) where () () () 11 31 21 22 32 21 33 32 31 23 23 22 33 21 31 13 13 12 11 33 32 12 12 13 23 22 11 00 0 00 0 00 0 0 0 0 ∂∂∂ ⎡ ⎤ ⎢ ⎥ ∂∂ ∂ ⎢ ⎥ ⎢ ⎥ ∂∂. ⊕ ϕ ϕϕϕϕϕ (30) where 1, 2 11 12 T 13 1, 2 22 3 1, 2 11 12 13 T 3 [1, 1, ,0,0,0] ()2 2 [1, 1, 0,0,0,0] , 4,5,6 2 ii cc c c cc c i λ λ −− ⎫ =× ⎪ −− + ⎪ ⎬ ⎪ =− == ⎪ ⎭ , ξ ϕ ϕϕ ( 31) The eigenelasticity